Udacity Data Science Nanodegree

Project 1: Writing a Data Scientist Blog Post

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Notebook 4 of 4: London Airbnb Data

Business Question 4: Predict average daily listing price

by Juanita Smith

• Introduction
• Data gathering
• Feature Engineering
• Univariate Exploration
• Bivariate Exploration
• Multivariate Exploration
• Clustering
• Preprocessing
• Modelling Step 1 - Setting a baseline with Ridge and small alpha
• Modelling Step 2 - Model improvement with RidgeCV
• Final evaluation using TEST data
• Further evaluation using unseen Airbnb with date stamp 06/09/2023
• Conclusions
• References
• Submission

Introduction

In this notebook, we will use linear regression modelling to answer the last business question:

Business Question 4: Predict average daily listing price

Knowing the impact of business rentals, host excellence and star ratings from notebooks 2 and 3, could we make an accurate average daily base price prediction ?

This notebook will use the cleaned files created by running notebook 1 and 2 in sequence. Refer to the README file for data download instructions and metadata descriptions.

As the project is predicting a continues variable, we will make use of linear regression algorithms. To use linear regression, below assumptions needs to be tested to build a model that will generalize well.

Linear Regression Assumptions:

• Target variable is normally distributed
• There should be a linear relationship between the target and the features
• No Multicollinearity - Independent features should have no relationship between them
• Normality of the Error Terms, meaning residuals should be normally distributed with mean as centre
• Homoscedasticity - the spread of the residuals should be consistent along the regression line.

# avoid memory leaks
import os
os.environ["OMP_NUM_THREADS"] = '1'

# import all packages
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns

# clear the garbage to free memory as we are working with huge datasets
import gc 
from sklearn.linear_model import RidgeCV, Ridge
from sklearn.model_selection import train_test_split
from sklearn.metrics import r2_score, mean_squared_error, PredictionErrorDisplay, median_absolute_error, \
    mean_absolute_error
from sklearn.preprocessing import StandardScaler

import scipy as sp
from patsy import dmatrices
from scipy import stats

from scipy.special import boxcox, inv_boxcox

from sklearn.impute import SimpleImputer
from sklearn.preprocessing import OneHotEncoder
from sklearn.feature_selection import VarianceThreshold  
from sklearn.compose import TransformedTargetRegressor, ColumnTransformer

import statsmodels.api as sm
from statsmodels.stats.outliers_influence import variance_inflation_factor

from sklearn.model_selection import cross_validate, RepeatedStratifiedKFold
from sklearn.pipeline import  Pipeline

from statsmodels.stats.diagnostic import normal_ad

from sklearn.mixture import GaussianMixture

from sklearn.feature_selection import mutual_info_regression

from pickle import dump

# Import custom modules
import project1_data_science_blog.src.visuals as vs

# set plots to be embedded inline
%matplotlib inline

# suppress matplotlib user warnings
import warnings
warnings.filterwarnings("ignore", category=UserWarning, module="matplotlib")
warnings.filterwarnings("ignore", category=UserWarning, module="sklearn")

# # use high resolution if this project is run on an apple device
%config InlineBackend.figure_format='retina'

# Make your Jupyter Notebook wider
from IPython.display import display, HTML
display(HTML('<style>.container { width:80% !important; }</style>'))

# environment settings
# display all columns and rows during visual inspection
pd.options.display.max_columns = 999
pd.options.display.max_rows = 999

# geospatial analysis
import geopandas as gpd
from shapely.geometry import Point
from pyproj import CRS

gpd.options.display_precision = 15

# set default plot formatting and colors
BASE_COLOR, BASE_HIGHLIGHT_INTENSE, BASE_HIGHLIGHT, BASE_COMPLIMENTARY, BASE_GREY, BASE_COLOR_SUB1, BASE_COLOR_SUB2, SYMBOLS = vs.set_plot_defaults()

# default file paths

# time stamp model was developed with
date_stamp = '060623'

# latest time stamp we can use to validate our model as an optional last step towards the end.
date_stamp_latest = '060923'

FILE_PATH_CLEAN = '../data/clean/'

FILE_NAME_ENHANCED = FILE_PATH_CLEAN + 'listing_enhanced_' + date_stamp + '.pkl'
FILE_NAME_ENHANCED_LATEST = FILE_PATH_CLEAN + 'listing_enhanced_' + date_stamp_latest + '.pkl'

random_state = 77

def log_transform(x):
    return np.log10(x)

def inverse_log_transform(x):
    return sp.special.exp10(x)

def goodness_of_fit(y, y_hat, show=False):
    """ 
    Calculates and returns the performance score between true and predicted values based on the metric chosen. 
    """

    #Calculate the performance scores
    r2 = r2_score(y, y_hat)
    mse = mean_squared_error(y, y_hat) 
    mae = mean_absolute_error(y, y_hat) 
    maeEd = median_absolute_error(y, y_hat) 

    if show:
        print("R-squared: {:.2f}".format(r2))
        print("Mean Squared Error (MSE): {:.2f}".format(mse))
        print("Median Absolute Error: £{:.2f} per night".format(maeEd))    
        print("Mean Absolute Error: £{:.2f} per night".format(mae))  
        

    return {
        "R-squared": f"{r2_score(y, y_hat):.3f}",
        "Median Absolute Error": f"£{median_absolute_error(y, y_hat):.2f} per night",
        "Mean Absolute Error": f"£{mean_absolute_error(y, y_hat):.2f} per night",
        "Mean Squared Error": f"{mean_squared_error(y, y_hat):.2f}",
    }  

def get_mi_score(X, y):
    mi = mutual_info_regression(X, y, random_state=10)
    mi = pd.Series(mi, index=X.columns).sort_values(ascending=False)
    return mi

def calculate_residuals(model, X, y):
    """
    Creates predictions on the features with the model and calculates residuals
    """
    predictions = model.predict(X)
    df_results = pd.DataFrame({'Actual': y, 'Predicted': predictions})
    df_results['Residuals'] = abs(df_results['Actual']) - abs(df_results['Predicted'])
    
    return df_results

def plot_model_performance_scatter(
    y, y_pred1, y_pred2, title1, title2, suptitle, kind='actual_vs_predicted', lower_lim=-200, upper_lim=1500, interval=100, figsize=(12,6)):
    """ 
    Compare the performance of 2 models plotting its predictions vs actual values
    """

    bins = np.arange(lower_lim, upper_lim+interval, interval)
    fig, (ax0, ax1) = plt.subplots(1, 2, sharey=True, figsize=figsize)

    plt.subplot(1,2,1)

    PredictionErrorDisplay.from_predictions(
        y,
        y_pred1,
        kind=kind,
        ax=ax0,
        scatter_kwargs={"alpha": 0.3, 'marker':'o', 'color':BASE_HIGHLIGHT_INTENSE, 's':50, 'edgecolor':'black', 'linewidths':0.3}
    )
    
    if kind == 'actual_vs_predicted':
        ax0.set_xticks(bins, bins, rotation=90)
        ax0.set_yticks(bins, bins)
        ax0.set_xlim(lower_lim,upper_lim)
        ax0.set_ylim(lower_lim,upper_lim)
    else:
        ax0.set_xticks(bins, bins, rotation=90)        
        ax0.set_xlim(lower_lim,upper_lim)        

    plt.subplot(1,2,2)
    PredictionErrorDisplay.from_predictions(
        y,
        y_pred2,
        kind=kind,
        ax=ax1,
        scatter_kwargs={"alpha": 0.3, 'marker':'o', 'color':BASE_HIGHLIGHT_INTENSE, 's':50, 'edgecolor':'black', 'linewidths':0.3},
    )
    
    if kind == 'actual_vs_predicted':
        ax1.set_xticks(bins, bins, rotation=90)
        ax1.set_yticks(bins, bins)
        ax1.set_xlim(lower_lim,upper_lim)
        ax1.set_ylim(lower_lim,upper_lim) 
        loc = 'upper left'  
    else:
        ax1.set_xticks(bins, bins, rotation=90)        
        ax1.set_xlim(lower_lim,upper_lim)
        loc = 'upper right'
        
    ax1.yaxis.set_tick_params(labelleft=True)

    # Add the score in the legend of each axis
    for ax, y_pred in zip([ax0, ax1], [y_pred1, y_pred2]):
        for name, score in goodness_of_fit(y, y_pred).items():
            ax.plot([], [], " ", label=f"{name}={score}")
        ax.legend(loc=loc)

    ax0.set_title(title1)
    ax1.set_title(title2)
    fig.suptitle(suptitle)
    plt.show()

def plot_model_performance_hist(y, y_pred1, y_pred2, title1=None, title2=None, suptitle=None, figsize=(12,12), price_max=1200, price_interval=25):
    """ Compare performance of 2 models by plotting actual vs predicted target with seaborn histogram """

    # Actual vs predicted price comparison
    fig, (ax1, ax2) = plt.subplots(2, 1, figsize=figsize, sharex=True, sharey=True)

    # create bins 
    xbins = np.arange(0, price_max+price_interval, price_interval)

    # plot for prediction from model 1
    ax1 = sns.histplot(y, bins=xbins, stat='percent', kde=False, color=BASE_COLOR, label='actual', ax=ax1)
    sns.histplot(y_pred1, bins=xbins, stat='percent', kde=False, color=BASE_COMPLIMENTARY, alpha=0.5, label='predicted', ax=ax1)
    ax1.set_title(title1)
    ax1.legend()

    # plot for prediction from model 2
    ax2 = sns.histplot(y, bins=xbins, stat='percent', kde=False, color=BASE_COLOR, label='actual', ax=ax2)
    sns.histplot(y_pred2, bins=xbins, stat='percent', kde=False, color=BASE_COMPLIMENTARY, alpha=0.5, label='predicted', ax=ax2)   
    ax2.set_title(title2)
    ax2.legend()
    
    plt.xticks(xbins, xbins, rotation=90)
    ax1.xaxis.set_tick_params(labelbottom=True, rotation=90)    
    plt.xlim(0,price_max)
    plt.suptitle(suptitle)  


    plt.show()

def normal_errors_assumption(model, features, label, ax, p_value_thresh=0.05, title=None):
    """
    Normality: Assumes that the error terms are normally distributed. 
    This means, the error term should have a mean of zero
    If they are not, nonlinear transformations of variables may solve this.
               
    This assumption being violated primarily causes issues with the confidence intervals
    
    p-value from the test - below 0.05 generally means non-normal
    """
    
    # Calculating residuals for the Anderson-Darling test
    df_results = calculate_residuals(model, features, label)

    # Performing the test on the residuals
    p_value = normal_ad(df_results['Residuals'])[1]
    
    # Reporting the normality of the residuals
    if p_value < p_value_thresh:
        text = 'Residuals are not normally distributed'
    else:
        text = 'Residuals are normally distributed'
    
    # Plotting the residuals distribution
    plt.title(title)
    
    ax = sns.histplot(df_results['Residuals'], bins=100, stat='density', kde=True, color=BASE_COLOR)
    ax = sns.kdeplot(df_results['Residuals'], color=BASE_HIGHLIGHT_INTENSE, ax=ax, linewidth=2)
              
    ax.plot([], [], " ", label=f"pvalue={p_value}: {text}")
    ax.legend(loc="upper left")

def distribution_of_errors(model1, model2, X, y, title1=None, title2=None, suptitle=None, alpha=None, figsize=(12,6), kind='kde'):
    """ compare normal distribution of errors for 2 models"""
    
    fig = plt.figure(figsize=figsize)
    
    if kind == 'kde':

        ax = plt.subplot(1,2,1)
        normal_errors_assumption(model=model1, features=X, label=y, ax=ax, title=title1) 

        ax2 = plt.subplot(1,2,2, sharey=ax)
        normal_errors_assumption(model=model2, features=X, label=y, ax=ax2, title=title2) 
        
    elif kind == 'hist':

        ax1 = plt.subplot(1,2,1)
        x = calculate_residuals(model1, X_train, y_train).values.flatten()
        plt.hist(x, bins=100, color=BASE_HIGHLIGHT_INTENSE)
        plt.xlim(-1000, 1000)
        ax1.set_title(title1)
        
        ax1.set_ylabel('Frequency')
        ax1.set_xlabel('Residuals')       

        ax2 = plt.subplot(1,2,2, sharex=ax1, sharey=ax1)
        x = calculate_residuals(model2, X_train, y_train).values.flatten()
        sns.histplot(x, bins=100, color=BASE_HIGHLIGHT_INTENSE)
        ax2.set_title(title2)

        ax2.set_ylabel('Frequency')
        ax2.set_xlabel('Residuals')
        
    plt.suptitle(suptitle, y=1.05)        
    plt.tight_layout()        

def homoscedasticity_assumption(model, X, y, ax):
    """
    Assumption 5: Homoscedasticity of Error Terms: 
    
    Assumes that the errors/residuals have a constant variance, with a mean of 0
    """
       
    # Calculating residuals for the plot
    df_results = calculate_residuals(model, X, y)
    df_results.reset_index(inplace=True)

    sns.scatterplot(x=df_results.index, y=df_results.Residuals, color=BASE_COLOR, marker='o', edgecolor='black', linewidth=0.1, alpha=0.2)
    plt.plot(np.repeat(0, df_results.index.max()), color=BASE_COMPLIMENTARY, linestyle='--')
    ax.spines['right'].set_visible(False)  # Removing the right spine
    ax.spines['top'].set_visible(False)  # Removing the top spine

def residual_variance(model1, model2, X, y, title1=None, title2=None, suptitle=None, figsize=(14,4)):
    """ Compare Homoscedasticityof 2 models """
    
    plt.figure(figsize=figsize)
    
    ax1 = plt.subplot(1,2,1)
    ax1.set_title(title1)
    ax1.set_xlabel('Index')    
    homoscedasticity_assumption(model1, X, y, ax=ax1) 


    ax2 = plt.subplot(1,2,2)
    ax2.set_title(title2)
    ax2.set_xlabel('Index')    
    homoscedasticity_assumption(model2, X, y, ax=ax2) 

    plt.suptitle(suptitle, y=1.05)
    plt.show()

def coef_weights(model, logscale=False):
    '''
    INPUT:
    coefficients - the coefficients of the linear model 
    X_train - the training data, so the column names can be used
    OUTPUT:
    coefs_df - a dataframe holding the coefficient, estimate, and abs(estimate)
    
    Provides a dataframe that can be used to understand the most influential coefficients
    in a linear model by providing the coefficient estimates along with the name of the 
    variable attached to the coefficient.
    '''
    coefs_df = pd.DataFrame()
    coefs_df['est_int'] = feature_names = model[:-1].get_feature_names_out()
    
    try:
        coefs_df['coefs'] = model[-1].coef_
        intercept = model[-1].intercept_
    except:
        coefs_df['coefs'] = model[-1].regressor_.coef_
        intercept = model[-1].regressor_.intercept_
        
    if logscale:
        intercept = inverse_log_transform(intercept)
        coefs_df['coefs'] = inverse_log_transform(coefs_df['coefs'])  

    coefs_df['abs_coefs'] = abs(coefs_df['coefs'])
    coefs_df = coefs_df.sort_values('abs_coefs', ascending=False)
    
    coefs_df = coefs_df.set_index('est_int')  
    
    return coefs_df, intercept

def plot_coef(model1, model2, title1, title2, suptitle, figsize=(12,20), logscale1=False, logscale2=False):
    
    # plot for model 1
    ax1 = plt.subplot(1,2,1)
    coef_df, intercept = coef_weights(model1, logscale1)
    coef_df['coefs'].plot.barh(figsize=figsize, color=BASE_COLOR, ax=ax1)
    plt.title(title1 + '\n Intercept: £{:.0f}'.format(intercept))
    start = 0
    plt.xlabel("Raw coefficient values")    
    if logscale1:
        start = 1    
    plt.axvline(x=start, color=".5")
    plt.ylabel('Independent features')
    plt.subplots_adjust(left=0.4)
    
    for c in ax1.containers:
        labels = [f"{x:,.3}" for x in c.datavalues]
        ax1.bar_label(c, label=labels, label_type='edge', fontsize=6, padding=3)   
    ax1.margins(x=0.2)    

    # plot for model 2
    ax2 = plt.subplot(1,2,2)
    coef_df, intercept = coef_weights(model2, logscale2)
    coef_df['coefs'].plot.barh(figsize=figsize, color=BASE_COLOR)
    plt.title(title2 + '\n Intercept: £{:.0f}'.format(intercept))
    plt.xlabel("Raw coefficient values")
    if logscale2:
        start = 1 
    plt.axvline(x=start, color=".5")
    plt.ylabel('Independent features')
    plt.subplots_adjust(left=0.4)
    
    for c in ax2.containers:
        labels = [f"{x:,.3}" for x in c.datavalues]
        ax2.bar_label(c, label=labels, label_type='edge', fontsize=6, padding=3)   
    ax2.margins(x=0.2)     

    plt.suptitle(suptitle, y=1.01)
    plt.tight_layout()
    
    return coef_df

def coefficient_variability(model, X, y, title, suptitle, splits=5, repeats=20, jobs=4, random_state=0, scoring='r2', figsize=(10,12)):

    feature_names = model[:-1].get_feature_names_out()

    cv = RepeatedStratifiedKFold(n_splits=splits, n_repeats=repeats, random_state=random_state)

    cv_model = cross_validate(
        model,
        X,
        y,
        scoring=scoring,
        cv=cv,
        return_estimator=True,
        n_jobs=jobs,
        return_train_score=True,
        verbose=0
    )

    try:
        coefs = pd.DataFrame(
            [est[-1].regressor_.coef_ for est in cv_model["estimator"]], columns=feature_names,
        )
    except:
        coefs = pd.DataFrame(
            [est[-1].coef_ for est in cv_model["estimator"]], columns=feature_names,
        )  
        
    plt.figure(figsize=figsize)
    sns.stripplot(data=coefs, orient="h", palette="dark:k", alpha=0.5)
    sns.boxplot(data=coefs, orient="h", color="cyan", saturation=0.5, whis=10)
    plt.axvline(x=0, color=".5")
    plt.xlabel("Coefficient importance")
    plt.title(title)
    suptitle = suptitle
    plt.suptitle(suptitle, y=1.01, x=0.65)
    plt.subplots_adjust(left=0.3)
    plt.tight_layout()
    
    return cv_model

def compare_qqplot(model1, model2, X, y, title1, title2, suptitle, figsize=(8,4)):
    """ Compare QQ plot for 2 models against normal distribution"""
    
    fig, (ax, ax2) = plt.subplots(1, 2, figsize=figsize, sharey=True, sharex=True)
    
    residuals = calculate_residuals(model1, X, y)
    qq = sm.qqplot(data=residuals['Residuals'], line='q', ax=ax, marker='.', markerfacecolor=BASE_COLOR, markeredgecolor=BASE_COLOR)    
    
    
    residuals = calculate_residuals(model2, X, y)
    sm.qqplot(data=residuals['Residuals'], line='q', ax=ax2, marker='.', markerfacecolor=BASE_COLOR, markeredgecolor=BASE_COLOR)
    
    ax.set_title(title1)
    ax2.set_title(title2)
    plt.suptitle(suptitle, y=1.05)
    plt.show()

# build a final cleaning function
def cleaning(df, cluster_preprocessor, cluster_model, onehot_preprocessor):

    # cluster preprocess for only relevant columns
    X_property = listing_latest[['room_type', 'property_type', 'latitude', 'longitude', 'price_mean']].copy(deep=True)
    X_property_transformed = cluster_preprocessor.transform(X_property) 
    
    # cluster data with already fitted model
    X_property_transformed['cluster'] = cluster_model.predict(X_property_transformed)
      
    # rank and renaming clusters to be in sequential order
    tmp = X_property_transformed.groupby('cluster').agg(min=('price_mean', 'min'), max=('price_mean', 'max'), nr_listings=('cluster', 'count')).sort_values(by='min')
    tmp['rank'] = tmp['min'].rank(method='max').astype('int') 
    
    # resequence the cluster names by price
    mymap = pd.Series(tmp['rank'].values,index=tmp.index).to_dict()    
    
    # rename clusters to sequence them into ranked order
    X_property_transformed['property_cluster'] = X_property_transformed['cluster'].map(mymap)  
    
    # add cluster to main dataset 
    df['property_cluster'] = X_property_transformed['property_cluster']    
    
    # drop lat/long after clustering
    df = df.drop(['latitude', 'longitude'], axis=1)
    
    # split the dataset into dependent and independent datasets
    y = df['price_mean'].copy()
    X = df.drop('price_mean', axis=1)  
    
    # final onehot encoding for all features dropping sparse columns
    X_transformed = onehot_preprocessor.transform(X)
    
    # make sure property types are well presented in both training and testing sets using stratify
    X_train, X_test, y_train, y_test = train_test_split(X_transformed, 
                                                        y, 
                                                        test_size=0.25, 
                                                        random_state=random_state,
                                                        stratify=X_transformed[['property_cluster', 
                                                                                'is_business_True']],
                                                       )
    
    return X_train, X_test, y_train, y_test

Gather Data

# features to be used in modelling
main_cols = ['host_is_superhost', 'accommodates', 'bedrooms', 'bathrooms',
        'minimum_minimum_nights', 'calculated_host_listings_count', 'host_years_active', 'price_mean',
        'private_bathroom', 'host_same_location', 'is_business', 'availability_90', 'neighbourhood_cleansed', 
        'property_type', 'room_type', 'review_scores_location', 'review_scores_value', 'latitude', 'longitude']

# load the enhanced listing file 
listing = pd.read_pickle(FILE_NAME_ENHANCED)
listing.head(3)

	listing_id	listing_url	name	host_id	host_name	host_response_time	host_response_rate	host_acceptance_rate	host_is_superhost	host_neighbourhood	host_has_profile_pic	host_identity_verified	neighbourhood_cleansed	latitude	longitude	property_type	room_type	accommodates	bathrooms	bedrooms	beds	price	minimum_minimum_nights	minimum_maximum_nights	availability_30	availability_60	availability_90	availability_365	number_of_reviews	number_of_reviews_ltm	number_of_reviews_l30d	first_review	last_review	review_scores_rating	review_scores_accuracy	review_scores_cleanliness	review_scores_checkin	review_scores_communication	review_scores_location	review_scores_value	instant_bookable	calculated_host_listings_count	calculated_host_listings_count_entire_homes	calculated_host_listings_count_private_rooms	calculated_host_listings_count_shared_rooms	reviews_per_month	host_has_about	host_has_neighborhood_overview	host_years_active	price_mean	private_bathroom	host_same_location	is_business	minimum_night_ranges
0	157714	https://www.airbnb.com/rooms/157714	Townhouse in Hammersmith · ★4.71 · 1 bedroom ·...	757377	Eilinn	within a few hours	90	68	False	Hammersmith	True	True	Hammersmith and Fulham	51.48974	-0.22208	Private room in townhouse	Private room	3	1.5	1.0	1.0	69	4	21	10	37	67	67	175	10	0	2011-07-13	2023-05-02	4.71	4.66	4.78	4.84	4.83	4.76	4.66	False	1	0	1	0	1.21	True	True	12	70	True	True	False	7
1	13913	https://www.airbnb.com/rooms/13913	Rental unit in Islington · ★4.80 · 1 bedroom ·...	54730	Alina	within an hour	100	78	False	LB of Islington	True	True	Islington	51.56861	-0.11270	Private room in rental unit	Private room	1	1.0	1.0	1.0	79	1	29	23	48	78	353	41	15	0	2010-08-18	2022-12-11	4.80	4.72	4.72	4.74	4.82	4.69	4.69	False	2	1	1	0	0.26	True	True	14	79	False	True	False	7
2	15400	https://www.airbnb.com/rooms/15400	Rental unit in London · ★4.80 · 1 bedroom · 1 ...	60302	Philippa	within a day	100	47	False	Chelsea	True	True	Kensington and Chelsea	51.48780	-0.16813	Entire rental unit	Entire home/apt	2	1.0	1.0	1.0	100	7	30	0	2	2	70	94	5	0	2009-12-21	2023-05-01	4.80	4.85	4.88	4.88	4.83	4.93	4.74	False	1	1	0	0	0.57	True	True	14	107	True	True	False	7

# get neighborhood geoshapes for spatial analysis
crs_4326 = CRS("WGS84")
neighbourhood = gpd.read_file('../data/raw/neighbourhood_geojson/london/neighbourhoods_060623.geojson', crs=crs_4326)
neighbourhood.head()

	neighbourhood	neighbourhood_group	geometry
0	Kingston upon Thames	None	MULTIPOLYGON (((-0.330679000000000 51.32901100...
1	Croydon	None	MULTIPOLYGON (((-0.064021000000000 51.31863800...
2	Bromley	None	MULTIPOLYGON (((0.012131000000000 51.299599000...
3	Hounslow	None	MULTIPOLYGON (((-0.244562000000000 51.48870200...
4	Ealing	None	MULTIPOLYGON (((-0.411833000000000 51.53408400...

neighbourhood.shape

(33, 3)

Feature engineering

listing_clean = listing.copy()

# make sure that each record is a unique listing  
listing_clean['listing_id'].duplicated().sum()

0

# for easy trouble shooting, lets make listing_id the index
listing_clean = listing_clean.set_index('listing_id')
listing_clean = listing_clean.loc[:, main_cols]
listing_clean.head()

	host_is_superhost	accommodates	bedrooms	bathrooms	minimum_minimum_nights	calculated_host_listings_count	host_years_active	price_mean	private_bathroom	host_same_location	is_business	availability_90	neighbourhood_cleansed	property_type	room_type	review_scores_location	review_scores_value	latitude	longitude
listing_id
157714	False	3	1.0	1.5	4	1	12	70	True	True	False	67	Hammersmith and Fulham	Private room in townhouse	Private room	4.76	4.66	51.48974	-0.22208
13913	False	1	1.0	1.0	1	2	14	79	False	True	False	78	Islington	Private room in rental unit	Private room	4.69	4.69	51.56861	-0.11270
15400	False	2	1.0	1.0	7	1	14	107	True	True	False	2	Kensington and Chelsea	Entire rental unit	Entire home/apt	4.93	4.74	51.48780	-0.16813
306333	False	5	2.0	1.0	2	1	11	198	True	False	False	4	Hackney	Entire rental unit	Entire home/apt	4.89	4.53	51.52748	-0.08172
159736	False	2	1.0	1.0	4	4	12	62	False	False	False	2	Lambeth	Private room in rental unit	Private room	4.34	4.66	51.46788	-0.09993

# make sure features that will be one hot encoded have no spaces in it's values, as it will become part of heading
categorical_columns = ['neighbourhood_cleansed', 'property_type', 'room_type']
listing_clean[categorical_columns] = listing_clean[categorical_columns].apply(lambda x: x.str.replace(' ', '_').str.lower())
listing_clean[categorical_columns] = listing_clean[categorical_columns].apply(lambda x: x.str.replace('/', '_').str.lower())

neighbourhood['neighbourhood'] = neighbourhood['neighbourhood'].apply(lambda x: x.replace(' ', '_').lower())
neighbourhood['neighbourhood'] = neighbourhood['neighbourhood'].apply(lambda x: x.replace('/', '_').lower())

# check for outliers
listing_clean.describe([0.025, 0.25, 0.5, 0.75, 0.9, 0.95, 0.975, 0.99, 0.999])

	accommodates	bedrooms	bathrooms	minimum_minimum_nights	calculated_host_listings_count	host_years_active	price_mean	availability_90	review_scores_location	review_scores_value	latitude	longitude
count	68923.000000	68867.000000	68842.000000	68923.000000	68923.000000	68923.000000	68923.000000	68923.000000	47720.000000	47719.000000	68923.000000	68923.000000
mean	3.168014	1.475975	1.338630	4.350202	19.274059	6.251208	202.492608	33.066770	4.709157	4.553347	51.509690	-0.129571
std	1.985088	0.962448	0.676587	9.065348	62.684541	3.230126	240.753573	31.683168	0.441736	0.564701	0.048800	0.101330
min	1.000000	0.000000	0.000000	1.000000	1.000000	0.000000	7.000000	0.000000	0.000000	0.000000	51.295937	-0.497800
2.5%	1.000000	0.000000	1.000000	1.000000	1.000000	0.000000	30.000000	0.000000	3.670000	3.000000	51.398398	-0.344240
25%	2.000000	1.000000	1.000000	1.000000	1.000000	4.000000	70.000000	0.000000	4.620000	4.430000	51.484790	-0.190558
50%	2.000000	1.000000	1.000000	2.000000	2.000000	7.000000	129.000000	26.000000	4.830000	4.700000	51.513730	-0.129850
75%	4.000000	2.000000	1.500000	4.000000	7.000000	9.000000	232.000000	61.000000	5.000000	4.900000	51.538970	-0.069360
90%	6.000000	3.000000	2.000000	7.000000	40.000000	10.000000	409.000000	83.000000	5.000000	5.000000	51.566208	-0.011270
95%	7.000000	3.000000	2.500000	14.000000	96.000000	11.000000	619.000000	89.000000	5.000000	5.000000	51.588314	0.032107
97.5%	8.000000	4.000000	3.000000	28.000000	224.000000	12.000000	990.000000	90.000000	5.000000	5.000000	51.603030	0.085716
99%	10.000000	5.000000	3.500000	45.000000	291.000000	12.000000	1507.780000	90.000000	5.000000	5.000000	51.624190	0.141005
99.9%	16.000000	7.000000	5.500000	90.000000	543.000000	14.000000	1791.078000	90.000000	5.000000	5.000000	51.660925	0.217254
max	16.000000	50.000000	50.000000	90.000000	543.000000	15.000000	2000.000000	90.000000	5.000000	5.000000	51.681502	0.288570

# remove final outliers
listing_clean = listing_clean[listing_clean['accommodates'] <= 8]
listing_clean = listing_clean[listing_clean['bedrooms'] <= 5]
listing_clean = listing_clean[listing_clean['bathrooms'] <= 4]
listing_clean = listing_clean[listing_clean['price_mean'].between(30, 1500)]
listing_clean.describe([0.025, 0.25, 0.5, 0.75, 0.9, 0.95, 0.975, 0.99, 0.999])

	accommodates	bedrooms	bathrooms	minimum_minimum_nights	calculated_host_listings_count	host_years_active	price_mean	availability_90	review_scores_location	review_scores_value	latitude	longitude
count	65181.000000	65181.000000	65181.000000	65181.000000	65181.000000	65181.000000	65181.000000	65181.000000	45741.000000	45741.000000	65181.000000	65181.000000
mean	3.069054	1.423175	1.300916	4.327473	16.692380	6.233703	186.209616	33.416210	4.714553	4.557691	51.509394	-0.130231
std	1.683950	0.819303	0.530518	9.078507	52.108377	3.255960	186.584035	31.544086	0.434996	0.557968	0.048391	0.100523
min	1.000000	0.000000	0.000000	1.000000	1.000000	0.000000	30.000000	0.000000	0.000000	0.000000	51.295937	-0.497800
2.5%	1.000000	0.000000	1.000000	1.000000	1.000000	0.000000	35.000000	0.000000	3.670000	3.000000	51.398645	-0.343115
25%	2.000000	1.000000	1.000000	1.000000	1.000000	4.000000	73.000000	0.000000	4.630000	4.430000	51.484850	-0.190697
50%	2.000000	1.000000	1.000000	2.000000	2.000000	7.000000	129.000000	27.000000	4.840000	4.710000	51.513620	-0.130720
75%	4.000000	2.000000	1.500000	4.000000	7.000000	9.000000	225.000000	61.000000	5.000000	4.900000	51.538110	-0.070180
90%	6.000000	3.000000	2.000000	7.000000	36.000000	10.000000	382.000000	83.000000	5.000000	5.000000	51.564930	-0.012780
95%	6.000000	3.000000	2.500000	14.000000	83.000000	11.000000	541.000000	89.000000	5.000000	5.000000	51.587160	0.029320
97.5%	7.000000	4.000000	3.000000	28.000000	177.000000	12.000000	752.000000	90.000000	5.000000	5.000000	51.602310	0.082670
99%	8.000000	4.000000	3.000000	45.000000	289.000000	12.000000	1000.000000	90.000000	5.000000	5.000000	51.623584	0.138979
99.9%	8.000000	5.000000	4.000000	90.000000	543.000000	14.000000	1412.000000	90.000000	5.000000	5.000000	51.660162	0.214168
max	8.000000	5.000000	4.000000	90.000000	543.000000	15.000000	1500.000000	90.000000	5.000000	5.000000	51.681502	0.278960

# correct datatypes
listing_clean['neighbourhood_cleansed'] = listing_clean['neighbourhood_cleansed'].astype('category')
listing_clean['room_type'] = listing_clean['room_type'].astype('category')
listing_clean['property_type'] = listing_clean['property_type'].astype('category')
listing_clean['bedrooms'] = listing_clean['bedrooms'].astype(int)

listing_clean.info()

<class 'pandas.core.frame.DataFrame'>
Index: 65181 entries, 157714 to 908453389034894825
Data columns (total 19 columns):
 #   Column                          Non-Null Count  Dtype   
---  ------                          --------------  -----   
 0   host_is_superhost               65181 non-null  bool    
 1   accommodates                    65181 non-null  int64   
 2   bedrooms                        65181 non-null  int64   
 3   bathrooms                       65181 non-null  float64 
 4   minimum_minimum_nights          65181 non-null  int64   
 5   calculated_host_listings_count  65181 non-null  int64   
 6   host_years_active               65181 non-null  int64   
 7   price_mean                      65181 non-null  int64   
 8   private_bathroom                65181 non-null  bool    
 9   host_same_location              65181 non-null  bool    
 10  is_business                     65181 non-null  bool    
 11  availability_90                 65181 non-null  int64   
 12  neighbourhood_cleansed          65181 non-null  category
 13  property_type                   65181 non-null  category
 14  room_type                       65181 non-null  category
 15  review_scores_location          45741 non-null  float64 
 16  review_scores_value             45741 non-null  float64 
 17  latitude                        65181 non-null  float64 
 18  longitude                       65181 non-null  float64 
dtypes: bool(4), category(3), float64(5), int64(7)
memory usage: 6.9 MB

Univariate Analysis

Let's double-check target feature distribution. Is it normally distributed?

Skew:

• If the skewness is between -0.5 and 0.5, the data are fairly symmetrical
• If the skewness is between -1 and — 0.5 or between 0.5 and 1, the data are moderately skewed
• If the skewness is less than -1 or greater than 1, the data are highly skewed

Kurtosis describes: the peakedness of the distribution.

• If the distribution is tall and thin it is called a leptokurtic distribution(Kurtosis > 3). Values in a leptokurtic distribution are near the mean or at the extremes.
• A flat distribution where the values are moderately spread out (i.e., unlike leptokurtic) is called platykurtic(Kurtosis <3) distribution.
• A distribution whose shape is in between a leptokurtic distribution and a platykurtic distribution is called a mesokurtic(Kurtosis=3) distribution. A mesokurtic distribution looks more close to a normal distribution.

plt.figure(figsize=(18,4))

plt.subplot(1,3,1)
xbinsize = 25
xbins = np.arange(0, listing_clean['price_mean'].max()+xbinsize, xbinsize)
ax1 = sns.histplot(data=listing_clean, x='price_mean', bins=xbins, kde=True, color=BASE_COLOR_SUB1)
plt.axvline(x=listing_clean['price_mean'].mean(), color='r', linestyle=':')
xticks=ax1.get_xticks()
labels = ['£{:1.0f}'.format(tick) for tick in xticks]
plt.xticks(xticks, labels)
plt.title('Untransformed distribution')
plt.xlabel('Mean price (per night)')
plt.ylabel('Frequency')
plt.subplot(1,3,2)

df = pd.DataFrame()
df['price_mean'], fitted_lambda = stats.boxcox(listing_clean['price_mean'])
ax2 = sns.histplot(data=df, x='price_mean', bins=100, color=BASE_COLOR_SUB2, kde=True)
plt.axvline(x=df['price_mean'].mean(), color='r', linestyle=':')
xticks=ax2.get_xticks()
labels = ['£{:1.0f}'.format(inv_boxcox(tick, fitted_lambda)) for tick in xticks]
plt.xticks(xticks,labels)
plt.title('Boxcox distribution')
plt.xlabel('Mean price (per night) in pounds')
plt.ylabel('Frequency')

plt.subplot(1,3,3)

df2 = pd.DataFrame()
df2['price_mean'] = log_transform(listing_clean['price_mean'])
ax3 = sns.histplot(data=df2, x='price_mean', bins=100, color=BASE_COLOR_SUB2, kde=True)
plt.axvline(x=df2['price_mean'].mean(), color='r', linestyle=':')
xticks=ax3.get_xticks()
labels = ['£{:1.0f}'.format(inverse_log_transform(tick)) for tick in xticks]
plt.xticks(xticks,labels)
plt.title('Log10 distribution')
plt.xlabel('Mean price (per night) in pounds')
plt.ylabel('Frequency')

plt.suptitle('Compare distributions of mean price per night (in pounds)')

# Add the skew and kurtosis score in the legend of each axis
for ax, df in zip([ax1, ax2, ax3], [listing_clean['price_mean'], df['price_mean'], df2['price_mean']]):
    skew = round(df.skew(),3)
    kurtosis = round(df.kurtosis(), 3)
    ax.plot([], [], " ", label=f"skew={skew} \nkurtosis={kurtosis}")
    ax.legend(loc='upper right')

plt.show()

Observation: Mean price is kind of normally distributed on both boxcox and log10 scale. Log10 scale seems to have higher density around the mean

Which transformation strategy to use ?

fig = plt.figure(figsize=(12,6))

ax1 = fig.add_subplot(221)
xt = listing_clean['price_mean']
stats.probplot(xt, dist=stats.norm, plot=ax1)
ax1.set_xlabel('')
ax1.set_title('Probability plot WITHOUT price transformation')
ax1.get_lines()[0].set_color(BASE_COLOR)
ax1.get_lines()[1].set_color(BASE_COMPLIMENTARY)

ax2 = fig.add_subplot(222)
xt, fitted_lambda = stats.boxcox(listing_clean['price_mean'])
stats.probplot(xt, dist=stats.norm, plot=ax2)
ax2.set_title('Probability plot WITH price BOXCOX transformation')
ax2.set_xlabel('')
yticks = ax2.get_yticks()
labels = ['{:1.0f}'.format(inv_boxcox(tick, fitted_lambda)) for tick in yticks]
plt.yticks(yticks,labels)
ax2.get_lines()[0].set_color(BASE_COLOR)
ax2.get_lines()[1].set_color(BASE_COMPLIMENTARY)

ax3 = fig.add_subplot(223)
xt = np.log1p(listing_clean['price_mean'])
stats.probplot(xt, dist=stats.norm, plot=ax3)
ax3.set_title('Probability plot WITH price LOG transformation')
yticks = ax3.get_yticks()
labels = ['{:1.0f}'.format(np.exp(tick)) for tick in yticks]
plt.yticks(yticks,labels)
ax3.get_lines()[0].set_color(BASE_COLOR)
ax3.get_lines()[1].set_color(BASE_COMPLIMENTARY)

ax4 = fig.add_subplot(224)
xt = log_transform(listing_clean['price_mean'])
stats.probplot(xt, dist=stats.norm, plot=ax4)
ax4.set_title('Probability plot WITH price LOG10 transformation')
yticks = ax4.get_yticks()
labels = ['{:1.0f}'.format(inverse_log_transform(tick)) for tick in yticks]
plt.yticks(yticks,labels)
ax4.get_lines()[0].set_color(BASE_COLOR)
ax4.get_lines()[1].set_color(BASE_COMPLIMENTARY)

plt.tight_layout()

Let's double-check the distribution of independent variables

# we want different approaches for numeric vs categorical features
numerical_features = listing_clean.select_dtypes(include='number').columns.tolist() 
numerical_features.remove('price_mean')
numerical_features.remove('latitude')
numerical_features.remove('longitude')
categorical_features = listing_clean.select_dtypes(exclude='number').columns.tolist() 

# distribution of numerical columns
pd.plotting.hist_frame(listing_clean, figsize=(20,8), bins=30, layout=(3,4), column=numerical_features, xlabelsize=8, ylabelsize=8, color=BASE_COLOR)
plt.suptitle('Distribution of numerical columns')
plt.show()

# plot distributions for categorical features
for col in categorical_features:
    vs.hist_by_cat(df=listing_clean, col=col, topn=30)

Bivariate Analysis

Correlation analysis

• Select features correlated with price
• Make sure independent variables does not have too high correlations with each other

df_corr = listing_clean.corr(method='spearman', numeric_only=True, min_periods=1)
sns.clustermap(df_corr, cmap="Spectral", center=0, annot=True, fmt='.2f', 
               annot_kws={"fontsize":8}, figsize=(10,10))

plt.title('Feature correlation')
plt.xticks(fontsize=10)
plt.show()

Which kind of relationship features have with price ?

#  Get the strongest correlation of features with mean price
corr_matrix = listing_clean[numerical_features].corrwith(listing_clean["price_mean"]).sort_values()

# Plot feature correlations to no_show
palette = sns.color_palette("viridis_r", 15)
pos = 'cadetblue'
neg = 'pink'
corr_matrix.plot(kind='bar', color=np.vectorize({True: pos, False: neg}.get)(corr_matrix > 0))
plt.xticks(rotation='vertical')
plt.ylabel('correlation')
plt.xlabel('features')
plt.title('Feature correlation with average listing price')
plt.show()

# Display relationship between price and numerical features
plt.figure(figsize=(14, 8))
for i, col in enumerate(numerical_features):
    plt.subplot(3, 3, i+1)
    sns.scatterplot(x=listing_clean[col], y=listing_clean['price_mean'], alpha=0.1, color=BASE_HIGHLIGHT_INTENSE, edgecolor='black', linewidth=0.05)
    plt.title(col)
    plt.xlabel(col)
    plt.ylabel('prices')
plt.suptitle('Relationship between price and numerical features', fontsize=14, y=1.05)
plt.tight_layout()

# Does features have a linear relationship with price ?

listing_clean['price_group'] = np.where(listing_clean['price_mean'] <= 500, 0, 1)
sns.set(style='ticks')
x_vars = numerical_features

for col in x_vars:
    ax = sns.lmplot(data=listing_clean, x=col, y='price_mean', hue='price_group', x_jitter=0.1, palette=[BASE_COLOR, BASE_COMPLIMENTARY], 
                    scatter_kws={'s':15, 'edgecolor':'black', 'linewidths':0.2}, height=3.5)

    ax = plt.gca()
    ax.set_title('{} price group trend'.format(col))    
    
plt.show()

# reset default plot formatting back as PairGrid change the defaults somehow
# BASE_COLOR, BASE_HIGHLIGHT_INTENSE, BASE_HIGHLIGHT, BASE_COMPLEMENTRY, BASE_GREY, BASE_COLOR_ARR, BASE_COLOR_DEP, SYMBOLS = vs.set_plot_defaults()

Observation: A linear relationship can be observed in most cases, however the slop of the fit changes for prices below and above £500, especially for number of bedrooms, bathrooms and number of people that can be accommodated.

Multivariate Analysis

Interaction analysis

Do features like neighbourhood and room type change the relationship some features have with price ?

# Interaction with neighbourhood
cat = listing_clean.groupby('neighbourhood_cleansed')['price_mean'].mean().sort_values(ascending=False)
hue_order = list(cat.index)


for col in ['bedrooms', 'bathrooms', 'accommodates', 'host_is_superhost', 'is_business']:          
    
    fig = plt.figure(figsize=(16,4))
    plt.subplot(1,2,1)
    sns.pointplot(data=listing_clean, x=col, y="price_mean",  color=BASE_COLOR)

    plt.subplot(1,2,2)
    sns.pointplot(data=listing_clean, x=col, y="price_mean", hue='neighbourhood_cleansed', hue_order=hue_order[:4], 
                  palette='Spectral_r')
    
    plt.suptitle('{} interaction with neighbourhood'.format(col))
    

plt.show()

/var/folders/4p/r9m3gynn3wj6h4yh4_lwz5fw0000gn/T/ipykernel_35437/1392799286.py:2: FutureWarning: The default of observed=False is deprecated and will be changed to True in a future version of pandas. Pass observed=False to retain current behavior or observed=True to adopt the future default and silence this warning.
  cat = listing_clean.groupby('neighbourhood_cleansed')['price_mean'].mean().sort_values(ascending=False)

Observation: There are indeed a lot of interactions introduced when we add features like neighbourhood and room type. Linear regression patterns can be observed, but the slope of the lines are changing for each neighbourhood

def custom(y, **kwargs):
    ym = y.mean()
    plt.axhline(ym, color=BASE_COMPLIMENTARY, linestyle="dashed")
    plt.annotate(f"mean: £{y.mean():.0f}", xy=(1,ym), 
                 xycoords=plt.gca().get_yaxis_transform(), ha="right", fontsize=12, color='r')


for col in ['neighbourhood_cleansed', 'room_type', 'property_type']:

    avg_price = listing_clean['price_mean'].mean()
    cat = listing_clean.groupby(col, observed=False)['price_mean'].mean().sort_values(ascending=False)
    order = list(cat.index)

    plt.figure(figsize=(16,6))
    ax = sns.pointplot(data=listing_clean, x=col, y='price_mean', errorbar='sd', marker='o', order=order, color=BASE_HIGHLIGHT_INTENSE)

    ax.axhline(avg_price, color=BASE_COMPLIMENTARY, linestyle="dashed", linewidth=2)    
    
    
    plt.title('Average price per night by {}'.format(col))
    plt.ylabel('Average price (per night) in pounds')
    plt.xlabel('col')

    plt.xticks(rotation=90)
    yticks = np.arange(0, 1300, 100)
    ylabels = ['£{:1.0f}'.format(tick) for tick in yticks] 
    plt.yticks(yticks, ylabels)

    plt.show()

Observation:

• Neighbourhood

• There's a constant variation in price between neighbourhoods, however more variation are observed in Westminster and Kensington and Chelsea

• Only the top 6 neighbourhoods show mean prices of above £200 per night, all other neighbourhoods have prices below £200

• Room type

• Entire homes and hotels show mean prices of above £200 per night, all other room types have prices below £200

• Property type

• Around 10 property types have extreme mean prices above £300 per night, with lots of variability like entire villas and townhouses

Let's explore a bit deeper the relationship between price and each type of room

def custom(y, **kwargs):
    ym = y.mean()
    plt.axhline(ym, color=BASE_COMPLIMENTARY, linestyle="dashed")
    plt.annotate(f"mean: £{y.mean():.0f}", xy=(1,ym), 
                 xycoords=plt.gca().get_yaxis_transform(), ha="right", fontsize=12, color='r', weight='bold')

g = sns.FacetGrid(data = listing_clean, row = 'room_type', height=5, aspect=3, sharex=True, sharey=True) 
g.map(sns.pointplot, 'property_type', 'price_mean', order=order, errorbar=('pi', 97.5), seed=88, linestyles=None, linewidth=2,
      err_kws={'linewidth': 1}, color=BASE_COLOR)

g = g.map(custom, 'price_mean')

plt.xticks(rotation=90)
plt.tight_layout()

Observation:

• Hotel rooms and entire homes or apartments show prices above £200 per night, whilst shared rooms and private rooms show prices below £100 per night

Let's use clustering to build a better price grouping

We have over 80 property types which could make the model after one-hot encoding too complex and cause over fitting. We could use room type instead, but this has only 4 categories with so much variation in price, it's too simple for modelling.

• Let's use both room type and property type, but cluster it for simplification.
• Let's minimize the interacts location, room type and property type brings, whilst getting rid of some of the relations between independent features like room type and property type

# encode selected categorical variables first before clustering

X_property = listing_clean[['room_type', 'property_type', 'latitude', 'longitude', 'price_mean']].copy(deep=True)

onehot_cluster = Pipeline(steps=[
    ('one-hot', OneHotEncoder(handle_unknown='ignore', sparse_output=False, dtype='int'))])

cluster_preprocessor = ColumnTransformer(transformers=[
    ('category', onehot_cluster, ['property_type', 'room_type']),
], 
        verbose_feature_names_out=False, 
        remainder = 'passthrough'                                                         
)

_ = cluster_preprocessor.fit(X_property).set_output(transform="pandas") 

X_property_transformed = cluster_preprocessor.transform(X_property)
X_property_transformed.head(2)

	property_type_barn	property_type_boat	property_type_camper_rv	property_type_campsite	property_type_casa_particular	property_type_castle	property_type_earthen_home	property_type_entire_bungalow	property_type_entire_cabin	property_type_entire_chalet	property_type_entire_condo	property_type_entire_cottage	property_type_entire_guest_suite	property_type_entire_guesthouse	property_type_entire_home	property_type_entire_home_apt	property_type_entire_loft	property_type_entire_place	property_type_entire_rental_unit	property_type_entire_serviced_apartment	property_type_entire_townhouse	property_type_entire_vacation_home	property_type_entire_villa	property_type_farm_stay	property_type_floor	property_type_houseboat	property_type_hut	property_type_island	property_type_private_room	property_type_private_room_in_bed_and_breakfast	property_type_private_room_in_boat	property_type_private_room_in_bungalow	property_type_private_room_in_cabin	property_type_private_room_in_camper_rv	property_type_private_room_in_casa_particular	property_type_private_room_in_castle	property_type_private_room_in_chalet	property_type_private_room_in_condo	property_type_private_room_in_cottage	property_type_private_room_in_earthen_home	property_type_private_room_in_farm_stay	property_type_private_room_in_floor	property_type_private_room_in_guest_suite	property_type_private_room_in_guesthouse	property_type_private_room_in_home	property_type_private_room_in_hostel	property_type_private_room_in_houseboat	property_type_private_room_in_hut	property_type_private_room_in_lighthouse	property_type_private_room_in_loft	property_type_private_room_in_minsu	property_type_private_room_in_nature_lodge	property_type_private_room_in_religious_building	property_type_private_room_in_rental_unit	property_type_private_room_in_serviced_apartment	property_type_private_room_in_shepherd's_hut	property_type_private_room_in_tiny_home	property_type_private_room_in_tower	property_type_private_room_in_townhouse	property_type_private_room_in_treehouse	property_type_private_room_in_vacation_home	property_type_private_room_in_villa	property_type_private_room_in_yurt	property_type_religious_building	property_type_riad	property_type_room_in_aparthotel	property_type_room_in_bed_and_breakfast	property_type_room_in_boutique_hotel	property_type_room_in_hostel	property_type_room_in_hotel	property_type_room_in_serviced_apartment	property_type_shared_room	property_type_shared_room_in_bed_and_breakfast	property_type_shared_room_in_boutique_hotel	property_type_shared_room_in_bungalow	property_type_shared_room_in_bus	property_type_shared_room_in_casa_particular	property_type_shared_room_in_condo	property_type_shared_room_in_earthen_home	property_type_shared_room_in_farm_stay	property_type_shared_room_in_guest_suite	property_type_shared_room_in_guesthouse	property_type_shared_room_in_home	property_type_shared_room_in_hostel	property_type_shared_room_in_hotel	property_type_shared_room_in_loft	property_type_shared_room_in_religious_building	property_type_shared_room_in_rental_unit	property_type_shared_room_in_serviced_apartment	property_type_shared_room_in_tent	property_type_shared_room_in_townhouse	property_type_shared_room_in_vacation_home	property_type_shared_room_in_villa	property_type_shepherd’s_hut	property_type_shipping_container	property_type_tiny_home	property_type_tower	property_type_yurt	room_type_entire_home_apt	room_type_hotel_room	room_type_private_room	room_type_shared_room	latitude	longitude	price_mean
listing_id
157714	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	51.48974	-0.22208	70
13913	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	51.56861	-0.11270	79

Let's use gausian mixture model (GMM) algorithm to do clustering, as it will look for clusters that have a normal price distribution within each cluster, which will provide a better split

# Clustering
gmm = GaussianMixture(n_components=25, random_state=random_state, covariance_type='spherical', init_params='k-means++')
X_property_transformed['cluster'] = gmm.fit_predict(X_property_transformed)
X_property_transformed.head(2)

	property_type_barn	property_type_boat	property_type_camper_rv	property_type_campsite	property_type_casa_particular	property_type_castle	property_type_earthen_home	property_type_entire_bungalow	property_type_entire_cabin	property_type_entire_chalet	property_type_entire_condo	property_type_entire_cottage	property_type_entire_guest_suite	property_type_entire_guesthouse	property_type_entire_home	property_type_entire_home_apt	property_type_entire_loft	property_type_entire_place	property_type_entire_rental_unit	property_type_entire_serviced_apartment	property_type_entire_townhouse	property_type_entire_vacation_home	property_type_entire_villa	property_type_farm_stay	property_type_floor	property_type_houseboat	property_type_hut	property_type_island	property_type_private_room	property_type_private_room_in_bed_and_breakfast	property_type_private_room_in_boat	property_type_private_room_in_bungalow	property_type_private_room_in_cabin	property_type_private_room_in_camper_rv	property_type_private_room_in_casa_particular	property_type_private_room_in_castle	property_type_private_room_in_chalet	property_type_private_room_in_condo	property_type_private_room_in_cottage	property_type_private_room_in_earthen_home	property_type_private_room_in_farm_stay	property_type_private_room_in_floor	property_type_private_room_in_guest_suite	property_type_private_room_in_guesthouse	property_type_private_room_in_home	property_type_private_room_in_hostel	property_type_private_room_in_houseboat	property_type_private_room_in_hut	property_type_private_room_in_lighthouse	property_type_private_room_in_loft	property_type_private_room_in_minsu	property_type_private_room_in_nature_lodge	property_type_private_room_in_religious_building	property_type_private_room_in_rental_unit	property_type_private_room_in_serviced_apartment	property_type_private_room_in_shepherd's_hut	property_type_private_room_in_tiny_home	property_type_private_room_in_tower	property_type_private_room_in_townhouse	property_type_private_room_in_treehouse	property_type_private_room_in_vacation_home	property_type_private_room_in_villa	property_type_private_room_in_yurt	property_type_religious_building	property_type_riad	property_type_room_in_aparthotel	property_type_room_in_bed_and_breakfast	property_type_room_in_boutique_hotel	property_type_room_in_hostel	property_type_room_in_hotel	property_type_room_in_serviced_apartment	property_type_shared_room	property_type_shared_room_in_bed_and_breakfast	property_type_shared_room_in_boutique_hotel	property_type_shared_room_in_bungalow	property_type_shared_room_in_bus	property_type_shared_room_in_casa_particular	property_type_shared_room_in_condo	property_type_shared_room_in_earthen_home	property_type_shared_room_in_farm_stay	property_type_shared_room_in_guest_suite	property_type_shared_room_in_guesthouse	property_type_shared_room_in_home	property_type_shared_room_in_hostel	property_type_shared_room_in_hotel	property_type_shared_room_in_loft	property_type_shared_room_in_religious_building	property_type_shared_room_in_rental_unit	property_type_shared_room_in_serviced_apartment	property_type_shared_room_in_tent	property_type_shared_room_in_townhouse	property_type_shared_room_in_vacation_home	property_type_shared_room_in_villa	property_type_shepherd’s_hut	property_type_shipping_container	property_type_tiny_home	property_type_tower	property_type_yurt	room_type_entire_home_apt	room_type_hotel_room	room_type_private_room	room_type_shared_room	latitude	longitude	price_mean	cluster
listing_id
157714	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	51.48974	-0.22208	70	0
13913	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	51.56861	-0.11270	79	16

# display the min/max prices for each cluster to check if the price groups make sense
tmp = X_property_transformed.groupby('cluster').agg(min=('price_mean', 'min'), max=('price_mean', 'max'), nr_listings=('cluster', 'count')).sort_values(by='min')
tmp['rank'] = tmp['min'].rank(method='max').astype('int')
tmp

	min	max	nr_listings	rank
cluster
17	30	47	6708	1
0	48	70	8896	2
16	71	91	7357	3
9	92	107	4771	4
22	108	126	4559	5
7	127	145	3813	6
18	146	172	5269	7
4	173	206	5592	8
14	207	236	3117	9
20	237	261	2501	10
2	262	290	1890	11
13	291	331	2352	12
8	332	400	2555	13
19	401	466	1373	14
5	467	516	904	15
23	517	572	621	16
10	573	630	563	17
1	631	722	567	18
12	723	817	395	19
3	819	921	313	20
24	922	1045	257	21
11	998	1001	365	22
15	1048	1250	243	23
6	1252	1412	135	24
21	1418	1500	65	25

# resquence the cluster names by price
mymap = pd.Series(tmp['rank'].values,index=tmp.index).to_dict()
mymap

{17: 1,
 0: 2,
 16: 3,
 9: 4,
 22: 5,
 7: 6,
 18: 7,
 4: 8,
 14: 9,
 20: 10,
 2: 11,
 13: 12,
 8: 13,
 19: 14,
 5: 15,
 23: 16,
 10: 17,
 1: 18,
 12: 19,
 3: 20,
 24: 21,
 11: 22,
 15: 23,
 6: 24,
 21: 25}

# rename clusters to sequence them into ranked order
X_property_transformed['property_cluster'] = X_property_transformed['cluster'].map(mymap)

# add cluster to main dataset and double check correctness
listing_clean['property_cluster'] = X_property_transformed['property_cluster']
listing_clean.head(3)

	host_is_superhost	accommodates	bedrooms	bathrooms	minimum_minimum_nights	calculated_host_listings_count	host_years_active	price_mean	private_bathroom	host_same_location	is_business	availability_90	neighbourhood_cleansed	property_type	room_type	review_scores_location	review_scores_value	latitude	longitude	price_group	property_cluster
listing_id
157714	False	3	1	1.5	4	1	12	70	True	True	False	67	hammersmith_and_fulham	private_room_in_townhouse	private_room	4.76	4.66	51.48974	-0.22208	0	2
13913	False	1	1	1.0	1	2	14	79	False	True	False	78	islington	private_room_in_rental_unit	private_room	4.69	4.69	51.56861	-0.11270	0	3
15400	False	2	1	1.0	7	1	14	107	True	True	False	2	kensington_and_chelsea	entire_rental_unit	entire_home_apt	4.93	4.74	51.48780	-0.16813	0	4

Let's use spatial analysis to visualize average prices in neighbourhoods more clearly

# make a geo spatial dataset with listing locations
cols = ['latitude', 'longitude', 'price_mean', 'property_cluster', 'neighbourhood_cleansed']
listings_geo = listing_clean[cols].copy(deep=True)
listings_geo.rename(columns={'neighbourhood_cleansed': 'neighbourhood'}, inplace=True)
listings_geo.head()

	latitude	longitude	price_mean	property_cluster	neighbourhood
listing_id
157714	51.48974	-0.22208	70	2	hammersmith_and_fulham
13913	51.56861	-0.11270	79	3	islington
15400	51.48780	-0.16813	107	4	kensington_and_chelsea
306333	51.52748	-0.08172	198	8	hackney
159736	51.46788	-0.09993	62	2	lambeth

# make geometry field
listings_geo['geometry'] = listings_geo.apply(lambda x: Point((x.longitude, x.latitude)), axis=1)
type(listings_geo)

pandas.core.frame.DataFrame

listings_geo = gpd.GeoDataFrame(listings_geo, crs=crs_4326, geometry=listings_geo.geometry)
listings_geo.head()

	latitude	longitude	price_mean	property_cluster	neighbourhood	geometry
listing_id
157714	51.48974	-0.22208	70	2	hammersmith_and_fulham	POINT (-0.222080000000000 51.489739999999998)
13913	51.56861	-0.11270	79	3	islington	POINT (-0.112700000000000 51.568610000000000)
15400	51.48780	-0.16813	107	4	kensington_and_chelsea	POINT (-0.168130000000000 51.487800000000000)
306333	51.52748	-0.08172	198	8	hackney	POINT (-0.081720000000000 51.527479999999997)
159736	51.46788	-0.09993	62	2	lambeth	POINT (-0.099930000000000 51.467880000000001)

# calculate average price per neighborhood
neigborhood_price = listings_geo.groupby('neighbourhood', as_index=False, observed=False)['price_mean'].mean()

neighbourhood = neighbourhood.merge(neigborhood_price, on='neighbourhood')
neighbourhood.head()

	neighbourhood	neighbourhood_group	geometry	price_mean
0	kingston_upon_thames	None	MULTIPOLYGON (((-0.330679000000000 51.32901100...	123.725490
1	croydon	None	MULTIPOLYGON (((-0.064021000000000 51.31863800...	100.060284
2	bromley	None	MULTIPOLYGON (((0.012131000000000 51.299599000...	114.579585
3	hounslow	None	MULTIPOLYGON (((-0.244562000000000 51.48870200...	135.320796
4	ealing	None	MULTIPOLYGON (((-0.411833000000000 51.53408400...	128.708951

fig, ax = plt.subplots(1, 1, figsize=(16,8))

lgnd_kwds = {'label': 'Average price per neighbourhood', 
             'orientation': 'vertical'}

base = neighbourhood.plot(ax=ax,
                        column='price_mean', 
                        edgecolor='black',
                        cmap='OrRd', 
                        alpha=0.5,
                        zorder=1,                       
                        legend=True, 
                        legend_kwds=lgnd_kwds)

base.set_axis_off()

plt.suptitle('Average price for neighbourhoods on Greater London', fontsize=12)
plt.show()

Observation:

• The closer we get to the centre of Greater London, the more expensive the listings get

Visualize the new property clusters

def spatial_mapping(data, color):
    price_min = data.price_mean.min()
    price_max = data.price_mean.max()
    ax=plt.gca()
    base = data.plot(ax=plt.gca(), zorder=1, marker='o', markersize=40, edgecolor="black", linewidth=1, color=BASE_HIGHLIGHT_INTENSE) 
    neighbourhood.plot(color='white', edgecolor='black', zorder=2, alpha=0.4, ax=base)
    base.set_axis_off()
    new_title = ax.get_title() + '\n £{} - £{}'.format(price_min, price_max)
    ax.set_title(new_title, fontsize='18')

g = sns.FacetGrid(data=listings_geo, col = 'property_cluster', col_order=[1, 8, 23], height=8, aspect=1)
g.map_dataframe(spatial_mapping)

plt.suptitle('Property price clusters of Greater London', fontsize=20)
plt.tight_layout()

Observation:

• We have perfect clusters. Cluster 1 contains budget accommodation which can be found all over greater London. As the cluster number increase, prices becomes more expensive, and listings are more condensed around the centre.

# analyse the relationship between the new property cluster and listing price
plt.figure(figsize=(20, 10))
for i, col in enumerate(['property_cluster']):
    plt.subplot(3, 3, i+1)
    sns.scatterplot(x=listing_clean[col], y=listing_clean['price_mean'], alpha=0.1, color=BASE_HIGHLIGHT_INTENSE, edgecolor='black', linewidth=0.05)
    plt.title('Relationship betweeen average price and {}'.format(col))
    plt.xlabel(col)
    plt.ylabel('prices')
plt.tight_layout()

Observation:

• The relationship are not perfect linear, this might cause some problems. It's a more curvy shape, which could lean towards polynomial regression.
• It's rare for properties to have prices over £500, it's a more luxury market, which might be the reason for the curve.

Model preprocessing

# clear the garbage to free memory
gc.collect()

249

Drop the fields no longer needed for modelling after clustering. Even though room type and location was used during clustering, we will still keep the room type and neighborhood features, to help the model identify the extreme budget and luxury fluctuations.

try:
    listing_clean = listing_clean.drop(['latitude', 'longitude', 'price_group'], axis=1)
except:
    pass

listing_clean.head(3)

	host_is_superhost	accommodates	bedrooms	bathrooms	minimum_minimum_nights	calculated_host_listings_count	host_years_active	price_mean	private_bathroom	host_same_location	is_business	availability_90	neighbourhood_cleansed	property_type	room_type	review_scores_location	review_scores_value	property_cluster
listing_id
157714	False	3	1	1.5	4	1	12	70	True	True	False	67	hammersmith_and_fulham	private_room_in_townhouse	private_room	4.76	4.66	2
13913	False	1	1	1.0	1	2	14	79	False	True	False	78	islington	private_room_in_rental_unit	private_room	4.69	4.69	3
15400	False	2	1	1.0	7	1	14	107	True	True	False	2	kensington_and_chelsea	entire_rental_unit	entire_home_apt	4.93	4.74	4

# split the datasets into dependent and independent datasets
y = listing_clean['price_mean'].copy()
X = listing_clean.drop('price_mean', axis=1)

# we want different preprocessing steps for numeric vs categorical features
numerical_features = X.select_dtypes(include='number').columns.tolist()   
print('Numerical features {}'.format(numerical_features))
categorical_features = X.select_dtypes(exclude='number').columns.tolist() 
print('Categorical features {}'.format(categorical_features)) 

Numerical features ['accommodates', 'bedrooms', 'bathrooms', 'minimum_minimum_nights', 'calculated_host_listings_count', 'host_years_active', 'availability_90', 'review_scores_location', 'review_scores_value', 'property_cluster']
Categorical features ['host_is_superhost', 'private_bathroom', 'host_same_location', 'is_business', 'neighbourhood_cleansed', 'property_type', 'room_type']

As we will use cross validation in steps below, we want to avoid that each fold end up with different features when we drop sparse columns after one-hot encoding. We therefore do only the onehot encoding step before modelling, using the full datasets to ensure consistency. As we now have a property cluster, we will only keep the top property types and get rid of exceptions to minimize interactions

# cleaning data / preprocessing step before we split out data into train/test
onehot_pipe = Pipeline(steps=[
    ('one-hot', OneHotEncoder(dtype='int', 
                              drop="if_binary", 
                              handle_unknown='infrequent_if_exist', 
                              sparse_output=False, 
                              min_frequency=40))   
])

onehot_preprocessor = ColumnTransformer(transformers=[
    ('category', onehot_pipe, categorical_features),
], 
        verbose_feature_names_out=False, 
        remainder = 'passthrough'                                                         
)

_ = onehot_preprocessor.fit(X).set_output(transform="pandas") 

X_transformed = onehot_preprocessor.transform(X)
X_transformed.head()

	host_is_superhost_True	private_bathroom_True	host_same_location_True	is_business_True	neighbourhood_cleansed_barking_and_dagenham	neighbourhood_cleansed_barnet	neighbourhood_cleansed_bexley	neighbourhood_cleansed_brent	neighbourhood_cleansed_bromley	neighbourhood_cleansed_camden	neighbourhood_cleansed_city_of_london	neighbourhood_cleansed_croydon	neighbourhood_cleansed_ealing	neighbourhood_cleansed_enfield	neighbourhood_cleansed_greenwich	neighbourhood_cleansed_hackney	neighbourhood_cleansed_hammersmith_and_fulham	neighbourhood_cleansed_haringey	neighbourhood_cleansed_harrow	neighbourhood_cleansed_havering	neighbourhood_cleansed_hillingdon	neighbourhood_cleansed_hounslow	neighbourhood_cleansed_islington	neighbourhood_cleansed_kensington_and_chelsea	neighbourhood_cleansed_kingston_upon_thames	neighbourhood_cleansed_lambeth	neighbourhood_cleansed_lewisham	neighbourhood_cleansed_merton	neighbourhood_cleansed_newham	neighbourhood_cleansed_redbridge	neighbourhood_cleansed_richmond_upon_thames	neighbourhood_cleansed_southwark	neighbourhood_cleansed_sutton	neighbourhood_cleansed_tower_hamlets	neighbourhood_cleansed_waltham_forest	neighbourhood_cleansed_wandsworth	neighbourhood_cleansed_westminster	property_type_entire_bungalow	property_type_entire_condo	property_type_entire_cottage	property_type_entire_guest_suite	property_type_entire_guesthouse	property_type_entire_home	property_type_entire_loft	property_type_entire_place	property_type_entire_rental_unit	property_type_entire_serviced_apartment	property_type_entire_townhouse	property_type_entire_vacation_home	property_type_private_room	property_type_private_room_in_bed_and_breakfast	property_type_private_room_in_bungalow	property_type_private_room_in_casa_particular	property_type_private_room_in_condo	property_type_private_room_in_guest_suite	property_type_private_room_in_guesthouse	property_type_private_room_in_home	property_type_private_room_in_hostel	property_type_private_room_in_loft	property_type_private_room_in_rental_unit	property_type_private_room_in_serviced_apartment	property_type_private_room_in_townhouse	property_type_private_room_in_vacation_home	property_type_room_in_aparthotel	property_type_room_in_boutique_hotel	property_type_room_in_hotel	property_type_room_in_serviced_apartment	property_type_shared_room_in_home	property_type_shared_room_in_rental_unit	property_type_infrequent_sklearn	room_type_entire_home_apt	room_type_hotel_room	room_type_private_room	room_type_shared_room	accommodates	bedrooms	bathrooms	minimum_minimum_nights	calculated_host_listings_count	host_years_active	availability_90	review_scores_location	review_scores_value	property_cluster
listing_id
157714	0	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	1	0	3	1	1.5	4	1	12	67	4.76	4.66	2
13913	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	1	0	1	1	1.0	1	2	14	78	4.69	4.69	3
15400	0	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	2	1	1.0	7	1	14	2	4.93	4.74	4
306333	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	5	2	1.0	2	1	11	4	4.89	4.53	8
159736	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	1	0	2	1	1.0	4	4	12	2	4.34	4.66	2

# make sure property types are well presented in both training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X_transformed, 
                                                    y, 
                                                    test_size=0.25, 
                                                    random_state=random_state,
                                                    stratify=X_transformed[['property_cluster', 'is_business_True']],
                                                   )

# pre-processing pipeline - scaling and impute data after splitting data into training and testing sets

numerical_preprocessor = Pipeline(steps=[
    ('impute', SimpleImputer(strategy='median')),
    ('scale', StandardScaler())
])

preprocessor = ColumnTransformer(transformers=[

    ('number', numerical_preprocessor, numerical_features),
], 
        verbose_feature_names_out=False, 
        remainder = 'passthrough'                                                         
)

Modelling Step 1 - Setting a baseline

2 baseline models will be created, 1 with target log transformation to make the distribution normal, and 1 without

As we are observed lots of interactions during EDA in the data when we introduce features neighbourhood and room_type, we will use the Ridge algorithm with regularization to minimize the impact

# As our base model, lets evaluate the performance of Ridge estimator with a very small alpha, with and without target log transformation

alpha_base=1e-10

# without target transformation
base_model = Pipeline(
    [
        ("preprocess", preprocessor),
        ('selector', VarianceThreshold(0)),
        ("base", Ridge(alpha_base)),                             
    ]
)

# with target transformation on log scale with base 10
base_model_with_log_target = Pipeline(
    [
        ("preprocess", preprocessor),
        ('selector', VarianceThreshold(0)),        
        ("base_transformed", TransformedTargetRegressor(regressor=Ridge(alpha_base), 
                                                        func=log_transform,
                                                        inverse_func=inverse_log_transform)),                     
    ]
)

_ = base_model.fit(X_train, y_train).set_output(transform="pandas")

_ = base_model_with_log_target.fit(X_train, y_train).set_output(transform="pandas")

Evaluation: base model

# predict
y_train_preds = base_model.predict(X_train)
y_train_preds_transformed = base_model_with_log_target.predict(X_train)

# transform X_train
X_train_transformed = preprocessor.transform(X_train)
X_train_transformed.head(3)

	accommodates	bedrooms	bathrooms	minimum_minimum_nights	calculated_host_listings_count	host_years_active	availability_90	review_scores_location	review_scores_value	property_cluster	host_is_superhost_True	private_bathroom_True	host_same_location_True	is_business_True	neighbourhood_cleansed_barking_and_dagenham	neighbourhood_cleansed_barnet	neighbourhood_cleansed_bexley	neighbourhood_cleansed_brent	neighbourhood_cleansed_bromley	neighbourhood_cleansed_camden	neighbourhood_cleansed_city_of_london	neighbourhood_cleansed_croydon	neighbourhood_cleansed_ealing	neighbourhood_cleansed_enfield	neighbourhood_cleansed_greenwich	neighbourhood_cleansed_hackney	neighbourhood_cleansed_hammersmith_and_fulham	neighbourhood_cleansed_haringey	neighbourhood_cleansed_harrow	neighbourhood_cleansed_havering	neighbourhood_cleansed_hillingdon	neighbourhood_cleansed_hounslow	neighbourhood_cleansed_islington	neighbourhood_cleansed_kensington_and_chelsea	neighbourhood_cleansed_kingston_upon_thames	neighbourhood_cleansed_lambeth	neighbourhood_cleansed_lewisham	neighbourhood_cleansed_merton	neighbourhood_cleansed_newham	neighbourhood_cleansed_redbridge	neighbourhood_cleansed_richmond_upon_thames	neighbourhood_cleansed_southwark	neighbourhood_cleansed_sutton	neighbourhood_cleansed_tower_hamlets	neighbourhood_cleansed_waltham_forest	neighbourhood_cleansed_wandsworth	neighbourhood_cleansed_westminster	property_type_entire_bungalow	property_type_entire_condo	property_type_entire_cottage	property_type_entire_guest_suite	property_type_entire_guesthouse	property_type_entire_home	property_type_entire_loft	property_type_entire_place	property_type_entire_rental_unit	property_type_entire_serviced_apartment	property_type_entire_townhouse	property_type_entire_vacation_home	property_type_private_room	property_type_private_room_in_bed_and_breakfast	property_type_private_room_in_bungalow	property_type_private_room_in_casa_particular	property_type_private_room_in_condo	property_type_private_room_in_guest_suite	property_type_private_room_in_guesthouse	property_type_private_room_in_home	property_type_private_room_in_hostel	property_type_private_room_in_loft	property_type_private_room_in_rental_unit	property_type_private_room_in_serviced_apartment	property_type_private_room_in_townhouse	property_type_private_room_in_vacation_home	property_type_room_in_aparthotel	property_type_room_in_boutique_hotel	property_type_room_in_hotel	property_type_room_in_serviced_apartment	property_type_shared_room_in_home	property_type_shared_room_in_rental_unit	property_type_infrequent_sklearn	room_type_entire_home_apt	room_type_hotel_room	room_type_private_room	room_type_shared_room
listing_id
899815776871362256	0.553869	0.704052	1.313731	-0.256894	-0.302636	0.851341	0.557023	0.237322	0.224832	0.295509	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0
50224111	0.553869	-0.517296	-0.568225	-0.367192	-0.302636	0.237242	1.413330	0.673890	-0.221122	0.088514	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0
46602756	-0.632473	0.704052	-0.568225	0.074002	-0.302636	-0.990957	-1.060445	0.482891	-0.836964	0.088514	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0

Mutual Information (MI) - Feature Importance

Mutual Information (MI) is a way to show how every feature interact with the target variable in this case 'price'. To interact means how a particular feature changes the target variable. Note: Mutual Information only works with numerical data

A higher value of MI means a closer connection between the feature and the target indicating feature importance for training the model. However, the lower the MI score like 0 indicates a weak connection between the feature and the target.

# display Mutual Information scores of the features
mi_score = get_mi_score(X_train_transformed, y_train)

# Show to 25 most important features
topn = 25
plt.figure(figsize=(6, 8))
ax = sns.barplot(y=mi_score.index[:topn], x=mi_score[:topn])
plt.xlabel('MI score')
ax.set_title('Mutual Information (MI) scores')
plt.show()

Test for multicollinearity using Variance Inflation Factor (VIF)

Multicollinearity means, there are linear relationships or correlation between independent variables, which should not be the case for linear regression to be effective.

Typically, a variable with VIF greater than 5 indicates Multicollinearity.

# gather features
vip_features = " + ".join(X_train_transformed.columns)

# merge X and y as required by statsmodels
vip_df = X_train_transformed.merge(y_train, left_index=True, right_index=True)

sm_y, sm_X = dmatrices('price_mean ~' + vip_features, vip_df, return_type='dataframe')
vif = pd.DataFrame()
vif['VIF Factor'] = [variance_inflation_factor(sm_X.values, i) for i in range(sm_X.shape[1])]
vif['features'] = sm_X.columns
vif

Observation:

• For VIF to be 'inf' means R2 must be 1. As the formula 1/1-R2, we end up dividing by 0, ending up with 'inf' values. It means we have high multicollinearity.
• Lots of interactions between neighborhood and room type exist, ideally we should drop 1 of these variables. But as we are using ridge algorithm, lets observe how regularization can take care of this problem.
• As accommodates, bedrooms and bathrooms have VIP below 5, I assume it's ok to keep all 3 variables, even though they have some correlation.

Plot model performance

plot_model_performance_scatter(
    y_train, 
    y_train_preds, 
    y_train_preds_transformed, 
    title1='WITHOUT target transformation', 
    title2='WITH target transformation', 
    suptitle='BASE MODEL Performance - Ridge regression (alpha = {})'.format(alpha_base),
    kind='actual_vs_predicted',  
    lower_lim=-100, 
    upper_lim=1500, 
    interval=100,
)

# Lets zoom a bit to the most likely price range, filtering the luxury listings out
plot_model_performance_scatter(
    y_train, 
    y_train_preds, 
    y_train_preds_transformed, 
    title1='WITHOUT target transformation', 
    title2='WITH target transformation', 
    suptitle='BASE MODEL Performance ZOOMED - Ridge regression (alpha = {})'.format(alpha_base),
    kind='actual_vs_predicted',  
    lower_lim=-25, 
    upper_lim=300, 
    interval=25,
)

Observation:

• Model without log transformation, are frequently under-predicting, and even comes up with negative price predictions. Overall prediction are better, which is quite strange ??!!
• Model with log transformation are doing excellent with most likely predictions between £100 - £500, but overpredict outlier budget listings below £75 and luxury listings over £500 which gives it poorer overall scores

plot_model_performance_hist(
    y_train, 
    y_train_preds, 
    y_train_preds_transformed, 
    title1='WITHOUT target transformation', 
    title2='WITH target transformation', 
    suptitle='BASE MODEL Actual vs Predicted - Ridge regression (alpha = {})'.format(alpha_base),
)

Observation: Log transformed model shows a better prediction exponential shape. Overall the model is doing great, but over fit on budget listings below £75

Linear Assumption Test: Normal distribution of residuals

Residuals should have a constant variance, with a mean around 0

distribution_of_errors(
    model1=base_model,
    model2=base_model_with_log_target,
    X=X_train,
    y=y_train,
    kind='kde',
    title1='WITHOUT target transformation', 
    title2='WITH target transformation', 
    suptitle='Distribution of residuals - Ridge regression with small regularization (alpha = {})'.format(alpha_base),
)

distribution_of_errors(
    model1=base_model,
    model2=base_model_with_log_target,
    X=X_train,
    y=y_train,
    kind='hist',
    title1='WITHOUT target transformation', 
    title2='WITH target transformation', 
    suptitle='Distribution of residuals - Ridge regression (alpha = {})'.format(alpha_base),
)

Observation: Log transformed model shows a bigger density around mean = 0, which is great, however the error range between 250 and -250 is bigger

Linear Assumption Test: Homoscedasticity

Residuals should have a constant variance, with a mean around 0

# Assumption #3: Homoscedasticity - residuals have a constant variance

plot_model_performance_scatter(
    y_train, 
    y_train_preds, 
    y_train_preds_transformed, 
    title1='WITHOUT target transformation', 
    title2='WITH target transformation', 
    suptitle='Linear Assumption Test: Homoscedasticity (alpha = {})'.format(alpha_base),
    kind='residual_vs_predicted',  
    lower_lim=0, 
    upper_lim=2000, 
    interval=100,    
)

residual_variance(
    model1=base_model,
    model2=base_model_with_log_target,
    X=X_train,
    y=y_train,
    title1='WITHOUT target transformation', 
    title2='WITH target transformation', 
    suptitle='Homoscedasticity of Error Terms (alpha = {})'.format(alpha_base),
    figsize=(14,4),
)

Observation: Both models have a constant variance for most listings, however there are in both cases some outliers. The variance is bigger where the model was log transformed

Inspect Model Coefficients

coef = plot_coef(
    model1=base_model,
    model2=base_model_with_log_target,
    title1='WITHOUT target transformation',
    title2='WITH target transformation',
    suptitle='Model coeffients - Ridge regression (alpha = {})'.format(alpha_base),
    figsize=(12,20),
    logscale1=False,
    logscale2=True,   
)

The plot above tells us about dependencies between a specific feature and the target when all other features remain constant, i.e., conditional dependencies.

Without log scale conversion, these coefficients represent actual pounds increase or decrease in price.

On log scale, these coefficients represent the % increase or decrease in price.

• For example 1.22 means a 22% increase in price for every 1 unit increase in that variable of all other variables stay constant.
• For example 0.8 means actually a 20% decrease in price for every 1 unit increase in that variable of all other variables stay constant.

Warning

Coeffients for model without log transformation makes little since. Intercept is high +/- £200 per night. An increase of ENTIRE HOMES will induce a decrease of the PRICE when all other features remain constant. On the contrary, an increase of the SHARED ROOMS will induce an increase of the PRICE when all other features remain constant. This contradicts what we saw in exploratory analysis above.

Coefficients for model with log transformation looks better. Intercept seems reasonable at around £115. We have some strange coefficients signs here as well. Example shared room in house would be expected to have coefficient value below 1.

Variability of the coefficients

We can check the coefficient variability through cross-validation: it is a form of data perturbation (related to resampling.

If coefficients vary significantly when changing the input dataset their robustness is not guaranteed, and they should probably be interpreted with caution.

cv_results_base_log = coefficient_variability(
    model=base_model_with_log_target, 
    X=X_train, 
    y=y_train, 
    splits=10,
    repeats=10,
    jobs=-1,
    scoring=('r2', 'neg_mean_squared_error'),
    figsize=(12,18),
    title='WITH target transformation',
    suptitle='BASE MODEL Coefficient variablility - Ridge model (alpha = {})'.format(alpha_base),
    random_state=random_state
)

# Print average scores during cross validation
print('Avg MSE score: {} with std {}'.format(cv_results_base_log['test_neg_mean_squared_error'].mean(), cv_results_base_log['test_neg_mean_squared_error'].std()))
print('Avg R2: {} with std {}'.format(cv_results_base_log['test_r2'].mean(), cv_results_base_log['test_r2'].std()))

Observation: We can observe high variability in room_type, and some minor variability in some neighbourhoods

QQ plot (short for Quantile-Quantile)

As the name suggests, it depicts the quantiles of the observed distribution (residuals in this case) against the quantiles of a reference distribution, typically the standard normal distribution.

• We plot the quantiles of the two distributions against each other.
• Deviations from the straight line indicate a departure from the assumed distribution.

A good QQ plot will:

Show minimal deviations from the reference line, indicating that the residuals are approximately normally distributed.

A bad QQ plot will:

Exhibit significant deviations, indicating a departure from the normality of residuals. Display patterns of skewness with its diverging ends, etc.

Thus, the more aligned the QQ plot looks, the more confident you can be about your model.

This is especially useful when the regression line is difficult to visualize, i.e., in a high-dimensional dataset.

#QQ plot
compare_qqplot(
    base_model, 
    base_model_with_log_target, 
    X_train, 
    y_train, 
    title1='WITHOUT target transformation', 
    title2='WITH target transformation', 
    suptitle='BASE model QQ plot - Ridge regression',
    figsize=(12,4)
)

Modelling Step 2 - Using RidgeCV

Let's get rid of interactions, by increasing regularization

from sklearn.model_selection import KFold

cv = KFold(n_splits=10, shuffle=True)

# manually set alpha quite high to see the effect.
alphas = (3000, 4000, 5000)

ridgecv = Pipeline(
    [
        ("preprocess", preprocessor),
        ('selector', VarianceThreshold(0)),
        ("regressor", TransformedTargetRegressor(regressor=RidgeCV(alphas=alphas, 
                                                                   scoring='neg_mean_squared_error', 
                                                                   cv=cv),
                                                func=log_transform,
                                                inverse_func=inverse_log_transform)),
    ]
)

ridgecv.fit(X_train, y_train).set_output(transform="pandas")  

First we check which value of $\alpha$ has been selected.

alpha_ridgecv = ridgecv.named_steps.regressor.regressor_.alpha_
print('alpha chosen: {}'.format(alpha_ridgecv))

# Best R2 score during fitting
print('Best R2 score during fitting: {}'.format(ridgecv.named_steps.regressor.regressor_.best_score_))

Evaluation: RidgeCV

def model_evaluation(model1, model2, X, y, alpha1=None, alpha2=None, title1=None, title2=None, suptitle=None):
      
    title1 = title1 + ' (alpha = {})'.format(alpha1)
    title2 = title2 + ' (alpha = {})'.format(alpha2) 
        
    # prediction
    y_pred_model1 = model1.predict(X)
    y_pred_model2 = model2.predict(X)    
    
    # model performance scatter plot: actual vs predicted
    plot_model_performance_scatter(
        y, 
        y_pred_model1, 
        y_pred_model2, 
        title1=title1,
        title2=title2,
        suptitle=suptitle + ' Model performance - Actual vs Predicted',
        kind='actual_vs_predicted',  
        lower_lim=-100, 
        upper_lim=2000, 
        interval=100,
        figsize=(12,6)
    )  
    
    # model performance scatter plot ZOOMED: actual vs predicted
    plot_model_performance_scatter(
        y, 
        y_pred_model1, 
        y_pred_model2, 
        title1=title1, 
        title2=title2, 
        suptitle=suptitle + ' Model performance - Actual vs Predicted',
        kind='actual_vs_predicted',  
        lower_lim=0, 
        upper_lim=300, 
        interval=25,
        figsize=(12,6)        
    )       
    
    # model performance histogram plot: actual vs predicted
    plot_model_performance_hist(
        y,
        y_pred_model1, 
        y_pred_model2, 
        title1=title1,
        title2=title2,
        suptitle=suptitle + ' Model performance - Actual vs Predicted',
        price_max=1500,
        price_interval=30,
        figsize=(12,12)
    )  
    
#   Linear Assumption Test: Distribution of Residuals'
    distribution_of_errors(
        model1=model1,
        model2=model2,
        X=X,
        y=y,
        kind='kde',
        title1=title1,
        title2=title2,
        suptitle=suptitle + ' Linear Assumption Test: Distribution of Residuals',
        figsize=(12,6)        
    )    
    
    distribution_of_errors(
        model1=model1,
        model2=model2,
        X=X,
        y=y,
        kind='hist',
        title1=title1,
        title2=title2,
        suptitle=suptitle + ' Linear Assumption Test: Distribution of Residuals',
        figsize=(12,6)        
    )    
    
    #  Linear Assumption Test: Homoscedasticity - residuals should have a constant variance with mean = 0
    plot_model_performance_scatter(
        y,
        y_pred_model1,
        y_pred_model2,
        title1=title1,
        title2=title2,
        suptitle=suptitle + ' Linear Assumption Test: Homoscedasticity of Error Terms',
        kind='residual_vs_predicted',  
        lower_lim=0, 
        upper_lim=1500, 
        interval=100,   
        figsize=(12,6)          
    )    
    
    residual_variance(
        model1=model1,
        model2=model2,
        X=X,
        y=y,
        title1=title1,
        title2=title2,
        suptitle=suptitle + ' Linear Assumption Test: Homoscedasticity of Error Terms',
        figsize=(14,4),
    )    

model_evaluation(
    model1=ridgecv, 
    model2=base_model_with_log_target, 
    X=X_train, 
    y=y_train,
    alpha1=alpha_ridgecv,
    alpha2=alpha_base,
    title1='RidgeCV',
    title2='Base Model - Ridge with log transformed target',
    suptitle='TRAIN Airbnb {}: '.format(date_stamp)
)

Inspect Model Coefficients

# on log scale, %
coef = plot_coef(
    model1=ridgecv,
    model2=base_model_with_log_target,
    title1='RidgeCV' + ' (alpha = {})'.format(alpha_ridgecv),
    title2='Base Model - Ridge with log transformed target' + ' (alpha = {})'.format(alpha_base),
    suptitle='Model Coefficents Importance',
    logscale1=True,
    logscale2=True,
    figsize=(12,20),
)

Observations:

On log scale, the coefficients contain the % increase in price, of all other variables stay constant. For example private_bathroom_True with of 1.082484 means for each increase in bathroom, price will go up by 8.24% if all other variables stay constant

Explanation how to interpret coefficients from Udacity mentor, I hope I got it right!

Intercept on Original Scale: The intercept value in your regression model is 2.12. When this is converted back to the original scale of the average price, it becomes approximately 130.83. This implies that if all other predictor variables are set to zero, the base predicted average price per day would be around £130.83.

Multiplier for accommodates Coefficient: The coefficient for the variable accommodates in your model is 0.0876. On the logarithmic scale, this coefficient signifies that for each unit increase in the accommodates variable, the log-transformed price is expected to increase by 0.0876. When this is translated back to the original price scale, it means that for each additional person accommodated, the predicted average price is multiplied by approximately 1.22. So, if accommodating one additional person, the price increases by about 22%.

I therefore interpret that these coefficients, when converted back, represent a percentage

Variability of the coefficients

cv_results = coefficient_variability(
    model=ridgecv, 
    X=X_train, 
    y=y_train, 
    splits=10,
    repeats=10,
    jobs=-1,
    scoring=('r2', 'neg_mean_squared_error'),
    figsize=(12,18),
    title='RidgeCV' + ' (alpha = {})'.format(alpha_ridgecv),
    suptitle='Coefficient importance and its variability',
    random_state=random_state
)

# Print average scores during cross validation
print('Avg MSE score: {} with std {}'.format(cv_results['test_neg_mean_squared_error'].mean(), cv_results['test_neg_mean_squared_error'].std()))
print('Avg R2: {} with std {}'.format(cv_results['test_r2'].mean(), cv_results['test_r2'].std()))

QQ plot (short for Quantile-Quantile)

#QQ plot
compare_qqplot(
    ridgecv, 
    base_model_with_log_target, 
    X_train, 
    y_train, 
    title1='RidgeCV' + ' (alpha = {})'.format(alpha_ridgecv),
    title2='Base Model - Ridge with log transformed target' + ' (alpha = {})'.format(alpha_base),
    suptitle='QQ plot',
    figsize=(12,4)
)

Final TEST Evaluation

model_evaluation(
    model1=ridgecv, 
    model2=base_model_with_log_target, 
    X=X_test, 
    y=y_test,
    alpha1=alpha_ridgecv,
    alpha2=alpha_base,
    title1='RidgeCV',
    title2='Base Model - Ridge with log transformed target',
    suptitle='TEST Airbnb {}: '.format(date_stamp)
)

Further evaluation with Airbnb 06/09/2023 data (optional)

As an extra step, use the latest Airbnb data released on 06 September 2023 to test the model performance. This data was never seen before during development of this model, we can see how the model generalize on a brand-new dataset! We will be using the already fitted model on June 2023 data.

Notebook 1 and 2 was run manually in sequence by changing the date stamp parameter from 060623 to 090623 A new enhanced file was generated with suffix '_090623' in data/clean folder

# read latest airbnb listings file
listing_latest = pd.read_pickle(FILE_NAME_ENHANCED_LATEST)
listing_latest.head(2)

# for easy troubleshooting, lets make listing_id the index
listing_latest = listing_latest.set_index('listing_id')
listing_latest = listing_latest.loc[:, main_cols]
listing_latest.head(2)

# cluster, clean and split data
X_train_060923, X_test_060923, y_train_060923, y_test_060923 = cleaning(listing_latest, cluster_preprocessor, gmm, onehot_preprocessor)

# evaluate latest airbnb file using models trained with June 2023 data
model_evaluation(
    model1=ridgecv, 
    model2=base_model_with_log_target, 
    X=X_train_060923, 
    y=y_train_060923,
    alpha1=alpha_ridgecv,
    alpha2=alpha_base,
    title1='RidgeCV',
    title2='Base Model - Ridge with log transformed target',
    suptitle='TEST Airbnb {}: '.format(date_stamp_latest)
)

# save fitted models
dump(cluster_preprocessor, open('../data/models/cluster_preprocessor.pkl', 'wb'))
dump(gmm, open('../data/models/gmm.pkl', 'wb'))
dump(onehot_preprocessor, open('../data/models/onehot_preprocessor.pkl', 'wb'))

Conclusion

Linear regression modelling was selected as the target variable, as average rental price per night, is a continues variable. The target variable was converted to a log10 scale to be normally distributed, to meet the linear regression criteria.

A successful linear regression data science model using algorithm 'RidgeCV' was build to predict prices with a median absolute error of only £11 per night. This means, prices are accurately predicted within a +/- £11 range. R2 score on the test dataset was 0.976, which means the model explains 98% of the variability of the data, which is totally amazing. RidgeCV was chosen to minimize interactions introduced when adding features neighbourhood, room type and property type. We need a high level regularization with alpha at 3000 to obtain the low level of errors achieved.

To reduce model complexity, property type was summarized through clusterng. We have over 80 property types, which could make the model after one-hot encoding too complex and cause over fitting. We could use room type instead, but this has only 4 categories with so much variation in price, it's too simple for modelling. Room type, Property type, location coordinates (latitude, longitude) and price was used to cluster the properties into price groups.

Gausian mixture model (GMM) algorithm was used for clustering, as it will look for clusters that have a normal price distribution within each cluster, which will provide a better split.

The average rental price in London does not easily exceed £500, however outliers up to £1500 was kept to observe model behaviour. These outliers can be removed for further improvement subsequently. Generally, the model prediction up to £500 seems excellent if one observe the model performance actual vs predicted, residuals are perfectly linear up to +/£250

As a last step, the model was evaluated with a new dataset with time stamp 06/09/2023, which became available subsequently. Outliers was not removed as during modelling, so the model has to cope with exceptional prices below £30 and above £1500. Mean absolute error was a little higher at £16 and R2 score was 0.956, still a great result considering outliers was not removed.

References

• Common pitfalls in the interpretation of coefficients of linear models
• Mixed-type imputation for IterativeImputer
• kaggle feature engineering lesson
• boston house prices
• linear regression assumptions
• Regularization of linear regression model
• pipeline for multiple predictors
• when to use gamma GLMS
• feature engineering - clustering with K-means
• using FACETGRID to plot geopandas maps
• QQ plots
• Interpreting skew and kurtosis

Submission

# convert notebook to html
os.system('jupyter nbconvert --to html 4_Price_Prediction_latest.ipynb')